I see three plausible choices for comparing groups when the dependent variable is skewed.  One possibility is to simply conduct the ANOVA; ANOVA is robust to violations of normality.  Exactly when the violation of normality is too much for ANOVA is not clear, but a rule of thumb is skew great than 1 is too much.  Since you're getting close to a large skew, a second possibility you might choose is to transform your data and conduct the ANOVA on the transformed DV.  With positive skew, you might choose logarithmic (with a constant added to each value if you have any zeros) or square root (if none of your DV are negative).  I added a link for an online stats calculator that will help you transform data and determine if you made it more normal.  A third possibility is to conduct a nonparametric test, so it will not assume that the distributions are normal.  The most popular choice is the Kruskal-Wallis test, but it assumes that the variance in each group is the same.  Hope this helps, Muayyad. ~ Kevin

http://vassarstats.net/trans1.html

Skewness value of 0.71 is still within the range of normality or at least the data does not depart from normality. Thus, the parametric testing procedure of One Way Anova is applicable.

Remember the skewness range form -3.0 (extremely skewed to the left) to 3.0 (extremely skewed to the right) and -1.0 to 1.0 is within the normality range.

Remember - the ANOVA assumes equal variance in each group as well...

It seems Kruskal_wallis is suitable.

I had went through a BMJ manual which mentions its not the skewness  and kurtosis but the critical value which is evaluated by dividing skewness and kurtosis by standard error

if the critical value comes in the range +/- 1.96 it is considered normal but beyond the +/-  1.96 indicates the variable is not normally distributed

Source :  Medical Statistics A Guide to Data Analysis and Critical Appraisals Chapter 32 Continuous variables : descriptive statistics pg 32 BMJ Publication

 Dear Dr Ahmad

For test normaility do you use kolmograph-smironof ?

Dear Malahat, thank you for your answers.

for me, I do more than one check such as asking the SPSS for skewness, boxplot, examining outliers with 3 SD and above, normal curve. 

No I did not use Kolmograph-smironof, is it available in SPSS?

The Kruskal–Wallis one-way analysis of variance by ranks (named after William Kruskal and W. Allen Wallis) is a non-parametric method for testing whether samples originate from the same distribution.

It is used for comparing two or more samples that are independent, and that may have different sample sizes, and extends the Mann–Whitney U test to more than two groups. The parametric equivalent of the Kruskal-Wallis test is the one-way analysis of variance (ANOVA).

When rejecting the null hypothesis of the Kruskal-Wallis test, then at least one sample stochastically dominates at least one other sample. The test does not identify where this stochastic dominance occurs or for how many pairs of groups stochastic dominance obtains.

Dunn's test would help analyze the specific sample pairs for stochastic dominance.

Since it is a non-parametric method, the Kruskal–Wallis test does not assume a normal distribution of the residuals, unlike the analogous one-way analysis of variance.

If the researcher can make the more stringent assumptions of an identically shaped and scaled distribution for all groups, except for any difference in medians, then the null hypothesis is that the medians of all groups are equal, and the alternative hypothesis is that at least one population median of one group is different from the population median of at least one other group.

In the attached article, they faced the same problem, but they compared two groups. Please see page 318 (evaluation of primary end points). They used the SPSS version 15. 

They used log transformations or root transformation in order to to normalized the data.

For not continuous variables, they used Wilcoxon rank sums test (non-parametric); which uses the rank of an observation rather than the observation value itself in the calculation of P value. The P value obtained from the Wilcoxon test is based on rank order rather than specific percentage value.

the Komogorov-Smirnov test was used to test equality of distributions before the Wilcoxon test  to verify the that distributions before and after the intervention were similar.

Article Impact of Protocol Watch on Compliance With the Surviving Se...

Since it is nonparametric,  Kruskal-Wallis test will be the one you need; however, I will use ANOVA to test the result as well.

Kolmogorov-Smirnov is availabe in SPSS, as well as the Kolmogorov-Smirnov test with Lilliefors correction. For small samples you can also use the Shapiro-Wilks test. The fromer test can be found under analyze-->non-parametric test, whereas the latter two can be found under analyze-->descriptives-->explore.

Edit: it seems as if the K-M test under non-parametric tests is also Lilliefors corrected, now. If I remember correctly, this was different in former versions of SPSS, I am using V 23 since a couple of weeks, but maybe I am wrong

In my opinion, first test the data normality with other methods such as Shapiro Wilks, etc. If it is not normalized again, you can use data conversion methods such as logarithmic conversion and so on.

  If the data were not normalized.In the next step If the groups are independent, use the Kruskal -Wallis test and if the groups are related, use the Fridman test.@